I believe that euthanasia is perfectly fine if permission is granted from the suffering man or woman.
I think Duo had a great comment "If one has a right to life, then one has a right to death"
Well said.
As for animals. I believe that the owners should look at the quality of life their pet is living. If they are ill or injured (a potential for full recovery) but can eat on it's own, and seems to enjoy itself at times, then I feel that it should be up to you. If you don't want to take care of a sick or injured dog, first look and see if someone else will. If not, then it's in your hands. But if the dog can't move, won't eat, and seems to be suffering. Then I think it's justified and that you should euthanize the animal.
Pretty soon I will put my dog to sleep. Other than being blind, she is getting alzheimers disease and gets stuck in corners. She urinates all over the place, and just doesn't seem to know what's going on at all. She can't enjoy herself. It's really sad. So I think euthaniasia is the best thing for her.